1. Field of the Invention
The present invention relates to an imaging method based on calculation of logical energy.
2. Discussion Of The Background
Imaging in two or three dimensions is traditionally used in industrial and medical applications, for example to make it possible to locate and characterize defects or obstacles in the interior of a liquid or solid medium by means of a nondestructive inspection method.
Under the English expression “pulse echo”, there are already known ultrasonic imaging techniques using an ultrasonic transmitter/receiver probe capable of sending an incident pulsed wave to the propagation medium to be inspected and of receiving the field reflected or diffracted by the propagation medium in response to the incident wave, in order to transmit it for analysis.
Under the expression “topological gradient calculation method”, there is already known an imaging method based on this technique, especially from the article entitled “Flaw imaging with ultrasound: the time domain topological gradient method”, published in the periodical “Review of Progress in Quantitative NonDestructive Evaluation Vol. 24A, pp. 859-866, 2004”, and from the article entitled “Time domain topological gradient and time reversal analogy: an inverse method for ultrasonic target detection”, which appeared in the periodical “Wave Motion vol. 42(1), pp. 31-52, 2005”. This method necessitates knowing the properties of a flawless reference medium, or in other words one that has identical composition, identical dimensions and identical physical structure as the part to be inspected but that with certainty contains no defects whatsoever, in contrast to the part to be inspected, which may possibly exhibit defects and/or in homogeneities.
This method determines a “cost function”, which evaluates the correlation between the data obtained for the reference medium and those measured on the medium to be inspected.
Starting from the reference medium, into which infinitesimal holes are introduced virtually and progressively, a “sensitivity analysis” (as it is known in English) is made of the cost function, in order to deduce therefrom modifications of the topology of the medium. The first stage of this analysis consists in solving a direct problem and what is known as an “adjunct” problem:                solving the direct problem consists in simulating the ultrasonic field u0 generated by the propagation of an ultrasonic wave in a predetermined zone of a flawless reference part; and        solving the adjunct problem consists in simulating the ultrasonic field v0 generated in this predetermined zone of the reference part by the propagation of an ultrasonic wave corresponding to an incident wave um−u0, where um is the measured signal returned by the medium to be inspected in response to a known incident wave.        
This analysis then consists in undertaking an asymptotic expansion of the “cost function” as a function of the topology of the medium. The first order term of this expansion gives the expression of a topological gradient as a function of the values of the u0 and v0 series. The most negative values of the gradient indicate where to insert the infinitesimal holes in order to cause the value of the “cost function” to become smaller and thus to make the modified topology tend toward that of the medium under inspection.
As used here, the term “hole” is a generic term that designates a zone exhibiting an abrupt contrast in elastic properties compared with the rest of the medium.
For reasons of simplification and visualization, the calculation of the topological gradient can be replaced in equivalent manner by the calculation of the corresponding topological energy. The image of the medium is then obtained by plotting the levels assumed by the topological energy, the defects being located at high values of this energy.